How would I recreate this warped grid-like pattern?
I've tried both Surface Defom and Lattice (as recommended by Susu). It looks like lattice is more convenient in your case as the cage can be simpler:

Create your tile, subidive it a bit, give it a Subdivision Surface modifier, array it, give it a Lattice modifier.

Create a lattice, scale it in Object mode so that it contains the tiles, give it a bit of Resolution in the Object Data panel, but you can keep it flat on the W axis with a value of 1. Create 2 shapekeys in the same panel:





Keep the second shapekey selected, go in Edit mode, enable the Proportional Editing option, move the vertices to deform the lattice, the object will deform as well. Once back in Object mode, play with the shapekey Value: